Math Problem Statement

Given the following transaction matrix, find the gross output to meet the final demand of 200 units of Agriculture and 800 units of Industry. Producing Sector: Agriculture and Industry with respective values in a matrix form.

Solution

The problem presented here involves finding the gross output needed to meet the final demand in a basic input-output model. Given the transaction matrix and the final demand for agriculture and industry, we will calculate the total gross output for each sector.

The table provided shows that:

  • x1x_1 = Total output for the Agriculture sector
  • x2x_2 = Total output for the Industry sector

The transaction matrix is:

300 & 600 \\ 400 & 1200 \\ \end{bmatrix}$$ Where: - 300 units of agriculture are used in the production of agriculture. - 600 units of agriculture are used in the production of industry. - 400 units of industry are used in the production of agriculture. - 1200 units of industry are used in the production of industry. The final demand vector is: $$\begin{bmatrix} 100 \\ 400 \\ \end{bmatrix}$$ The equation to calculate the total output of the sectors can be written as: $$x_1 = 300 + 600 + 100 = 1000 \quad \text{(Agriculture sector total output)}$$ $$x_2 = 400 + 1200 + 400 = 2000 \quad \text{(Industry sector total output)}$$ Therefore, the total gross output required to meet the final demand of 200 units of agriculture and 800 units of industry is: $$x_1 = 1000, \quad x_2 = 2000$$ Would you like further explanation or details? --- Here are five related questions: 1. How do input-output models contribute to understanding interdependence in sectors? 2. What are the limitations of this type of input-output analysis? 3. How would the total output change if the final demand increased? 4. How can this model be applied to a real economy's sectors? 5. What other mathematical methods are used in economics to measure sectoral output? **Tip:** When working with input-output matrices, make sure to correctly account for all intermediate and final demands to get accurate sector outputs.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Input-Output Analysis
Matrix Multiplication

Formulas

Gross Output = (I - A)^(-1) * Final Demand
where I is the identity matrix, A is the transaction matrix, and Final Demand is the given demand vector.

Theorems

Leontief Input-Output Model

Suitable Grade Level

College Level

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